1. Introduction
A temporal graph
$G = (V,E,\pi )$
is a graph
$G=(V,E)$
together with an ordering
$\pi \,:\,E \to \{1,\ldots ,|E|\}$
on the edge set, interpreted as the times where the edges appear in the graph, often called the time stamps of
$G$
. We say that an edge
$e \in E$
precedes an edge
$e' \in E$
if
$\pi (e) \lt \pi (e')$
. A path from
$u$
to
$v$
is called
${increasing}$
if each edge used in the path precedes the edge that is used after it, and we say that
$v$
is reachable from
$u$
(or that
$u$
can reach
$v$
) if an increasing path exists from
$u$
to
$v$
. A set of vertices
$S \subseteq V$
is called a temporal clique if for all distinct vertices
$u,v \in S$
, there is an increasing path from
$u$
to
$v$
(and vice versa).
In this note we discuss temporal graphs where
$\pi$
is a uniform permutation on the edges and
$G$
is an Erdős-Rényi random graph. The resulting temporal graph is called a random simple temporal graph; RSTG for short. Motivated by modelling time-dependent propagation processes, this model was introduced by Casteigts, Raskin, Renken, and Zamaraev [Reference Casteigts, Raskin, Renken and Zamaraev5].
One may generate RSTGs by a simple method: start with the complete graph
$K_n$
, then assign each edge an independent
${\operatorname {uniform}}(0,1)$
random variable
$(U_e \,:\, e \in E)$
and delete every edge with
$U_e \gt p$
. In this construction, we say that
$e$
precedes
$e'$
if
$U_e \lt U_{e'}$
. We also call the labels
$U_e$
the time stamps. Importantly, creating i.i.d. uniform time stamps like this allows us to extend the notion of a temporal graph to infinite graphs which is needed for our analysis.
Casteigts, Raskin, Renken, and Zamaraev [Reference Casteigts, Raskin, Renken and Zamaraev5] studied connectivity properties of RSTGs. They identified the thresholds for different strengths of connectivity to be in the region where
$p = \frac {c\log (n)}{n}$
for some constant
$c \gt 0$
(Throughout the paper,
$\log$
denotes natural logarithm). Furthering this work, Broutin, Kamčev and Lugosi [Reference Broutin, Kamčev and Lugosi4] identified the asymptotic lengths of the longest and shortest increasing paths in RSTGs with high probability for values of
$p$
in this range. (We say that an event
$E=E(n)$
happens with high probability if
$\mathbb{P}(E) \to 1$
as
$n \to \infty$
). Becker, Casteigts, Crescenzi, Kodric, Renken, Raskin, and Zamaraev [Reference Becker, Casteigts and Crescenzi2] identified
$p = \frac {\log (n)}{n}$
as the threshold for the appearance of large temporal cliques. In particular, they showed that for every
$\epsilon \gt 0$
, when
$p \ge \frac {(1+\epsilon )\log (n)}{n}$
, then there is a temporal clique of size
$n-o(n)$
with high probability, while if
$p \le \frac {(1-\epsilon )\log (n)}{n}$
, then every temporal clique is of size
$o(n)$
with high probability.
RSTGs are a natural way to model time-dependent processes on networks like social interactions and infection spread. A closely related model is the random gossip protocol model, in which a sequence of edges
$e_1,\ldots ,e_k$
are chosen uniformly from the edges of
$K_n$
and constructed a graph
$G_{n,k}$
. Increasing paths are defined as for temporal graphs. Papers studying this model include Moon [Reference Moon11], Boyd and Steele [Reference Boyd and Steele3] and Haigh [Reference Haigh9].
For deterministic temporal graph models with random time stamps, see Chvátal and Komlós [Reference Chvátal and Komlós6], and Graham and Kleitman [Reference Graham and Kleitman8], Lavrov and Loh [Reference Lavrov and Loh10] and Angel, Ferber, Sudakov, and Tassion [Reference Angel, Ferber, Sudakov and Tassion1].
Our contribution is summarised in the following theorem. It shows that in the subcritical regime
$p= c\log n/n$
with
$c\lt 1$
, the size of the largest temporal clique is not only
$o(n)$
but in fact, of size
$O(1)$
, improving the upper bound of [Reference Becker, Casteigts and Crescenzi2]. This reveals a quite spectacular phase transition around
$p= \log n/n$
, since for
$c\gt 1$
, there is a temporal clique of size
$n-o(n)$
. The behaviour of the size of the largest temporal clique near the critical regime remains an intriguing research problem.
Theorem 1.
Let
$p = \frac {c\log (n)}{n}$
, and let
$G$
be an RSTG with edge probability
$p$
. If
$c \in (0,1)$
, then the largest temporal clique in
$G$
is of size at most
$\lceil \frac {1}{1-c}+1 \rceil$
with high probability.
Note that for
$c \leq \frac {1}{2}$
, Theorem1 asserts that
$G$
has no temporal clique of size
$4$
. This bound can’t be improved, since for
$p = \omega (\frac {1}{n})$
the static Erdős-Rényi graph contains a triangle with high probability. Moreover, every triangle is trivially a temporal clique of size
$3$
. We conjecture that the upper bound of Theorem1 is sharp for all
$c\in (0,1)$
.
The proof of Theorem1 is based on relating the number of vertices that are reachable by monotone paths from a typical vertex to the total progeny of a certain “temporal” branching process. We utilise the temporal branching process bounds to assert that the number of vertices that a collection of
$m \geq \lceil 1 + \frac {1}{1-c}\rceil$
vertices can reach is small enough so that the chance of them forming a component unlikely enough that expected number of components of size
$m$
tends to zero.
2. Temporal branching processes
We begin this section by introducing temporal branching processes that are the key tool in the proof of Theorem1. We only focus on branching processes with binomial offspring distribution as this is the degree distribution of a typical vertex in an Erdős-Rényi random graph. One way to generate these processes is as follows. Start with an infinite rooted
$n$
-ary tree, add an independent
${\operatorname {uniform}}(0,1)$
time stamp
$U_e$
to every edge. Delete any edge with
$U_e \gt p$
. This decomposes the tree into a forest and we only focus on the component that contains the root vertex. We say that a vertex
$v$
is reachable from the root if the unique path from the root to
$v$
in the infinite
$n$
-ary tree has edge labels
$U_e \leq p$
for each edge on the path and these labels are increasing on the path. The subtree consisting of only vertices reachable from the root is a temporal branching process with a
${\operatorname {binomial}}(n,p)$
offspring distribution. Throughout the rest of the paper,
$T$
is always an infinite
$n$
-ary tree with such a labelling on the edges. The following sequence of results provides us with the necessary upper bounds for the size of the reachable set of
$T$
.
Lemma 2.
Let
$P_1,\ldots ,P_q$
be a finite collection of distinct infinite paths in
$T$
, and let
$(X_k)_{k \geq 0}$
be a random walk down the tree, that is,
$X_0$
is the root and
$X_k$
is uniformly distributed over the children of
$X_{k-1}$
for all
$k \geq 1$
. Then,
$\mathbb{P}(\tau \geq \ell ) \leq \frac {q}{n^\ell }$
, where
\begin{equation*}\tau = \max \{ k \gt 0 \,:\, X_0,\ldots ,X_k \text{ coincides with one of the $q$ paths}\}.\end{equation*}
Proof.
If
$\tau \geq \ell$
, then
$X_0,\ldots ,X_\ell$
coincides with one of the
$P_1,\ldots ,P_q$
. Since
$X_k$
is uniform over the children of
$X_{k-1}$
and the tree is
$n$
-ary, the result follows from the union bound yields.
Lemma 3.
Let
$T^*$
be the set of all vertices that are reachable from the root of
$T$
. If
$v_1,\ldots ,v_q$
are uniform vertices chosen from the
$\ell$
-th generation of
$T$
, then
\begin{equation*}\mathbb{P}(v_1,\ldots ,v_q \in T^*) \leq \frac {(q-1)!e^{npq}}{n^{\ell q}}.\end{equation*}
Furthermore, when
$\ell \geq (np)^4$
and
$np \to \infty$
as
$n \to \infty$
,
\begin{equation*}\mathbb{P}(v_1,\ldots ,v_q \in T^*) = O\left (\frac {(q-1)!C^{q-1}}{n^{\ell q}}\right ),\end{equation*}
for some
$C \gt 0$
.
Proof.
Suppose that
$v_1,\ldots ,v_r \in T^*$
and let
$T'$
be a subtree of
$T$
consisting of
$r$
distinct infinite paths starting at the root through
$v_1,\ldots ,v_r$
. Let
$(X_k)_{k \geq 0}$
be a random walk down the tree (independent of
$v_1,\ldots ,v_r$
) and let
$\tau$
be as in Lemma 2. If
$\tau = j$
, then there are
$\ell -j$
edges left that need to both exist and be increasing to have
$X_\ell \in T^*$
. Hence, if
$V_r = \{v_1 , \ldots , v_r \in T^*\}$
,
\begin{align*}\mathbb{P}(X_\ell \in T^* | V_r \cap \{\tau = j\}) &= p^{\ell -j}\mathbb{P}\big (\big \{\text{a path of length $\ell $ is increasing}\big \} \big |\big \{ \text{first $j$ edges are increasing}\big \}\big ) \\ &= \frac {p^{\ell -j}j!}{\ell !}.\end{align*}
Combining this with Lemma 2 yields
\begin{align} \mathbb{P}(X_\ell \in T^* | V_r) & \leq \sum _{j=0}^{\ell -1} \mathbb{P}(\tau = j)\mathbb{P}(X_\ell \in T^* | V_r \cap \{\tau = j\}) + \mathbb{P}(\tau \ge \ell )\mathbb{P}(X_\ell \in T^* | V_r \cap \{\tau \ge \ell \}) \nonumber \\ & \leq \sum _{j=0}^\ell \left (\frac {r}{n^j}\right )\frac {p^{\ell -j}j!}{\ell !} = \frac {r}{n^\ell }\sum _{j=0}^\ell \frac {(np)^{\ell -j}j!}{\ell !}. \end{align}
To get the first bound we use the fact that
$\frac {j!}{\ell !} \leq \frac {1}{(\ell -j)!}$
to get
\begin{equation*}\mathbb{P}(X_\ell \in T^*|V_r) \leq \frac {re^{np}}{n^\ell }.\end{equation*}
Then, since
$X_\ell$
is distributed uniformly across the
$\ell$
-th generation, applying the above inequality repeatedly,
\begin{align*} \mathbb{P}(v_1,\ldots ,v_q \in T^*) &= \mathbb{P}(v_1 \in T^*) \cdot \mathbb{P}(v_2 \in T^* | V_1) \cdots \mathbb{P}(v_q \in T^* | V_{q-1}) \\ &\leq \frac {p^\ell }{\ell !}\prod _{r=1}^{q-1}\left (\frac {re^{np}}{n^\ell }\right ) = \left (\frac {(np)^{\ell }}{\ell !}\right )\frac {(q-1)!e^{np(q-1)}}{n^{\ell q}} \leq \frac {(q-1)!e^{npq}}{n^{\ell q}}. \end{align*}
For the second inequality, we split the sum in (1) in two separate pieces
\begin{align*} A = \sum _{j=0}^{\ell - \sqrt {\ell }}\frac {(np)^{\ell - j}j!}{\ell !}, \quad \text{and} \quad B = \sum _{j=\ell -\sqrt {\ell }+1}^\ell \frac {(np)^{\ell -j}j!}{\ell !} = \sum _{k=0}^{\sqrt {\ell }-1} \frac {(np)^k}{\ell \cdot (\ell -1) \cdots (\ell -k+1)}. \end{align*}
The first term may be bounded by
\begin{align*} A &\leq \sum _{k=\sqrt {\ell }}^\ell \frac {(np)^k}{k!} \leq \left |e^{np}- \sum _{k=0}^{\sqrt {\ell }-1} \frac {(np)^k}{k!}\right | \leq \frac {e^{np}(np)^{\sqrt {\ell }}}{(\sqrt {\ell })!}\,, \end{align*}
where the second inequality follows from the Lagrange form of the remainder in Taylor’s theorem.
For the second term, since
$\ell \cdot (\ell -1) \cdots (\ell - k + 1) \geq \ell ^{k}(1-\frac {1}{\sqrt {\ell }})^{\sqrt {\ell }}$
for all
$0 \leq k \leq \sqrt {\ell }$
, we have that
\begin{align*} B &\leq \frac {1}{\left (1-\frac {1}{\sqrt {\ell }}\right )^{\sqrt {\ell }}}\sum _{k=0}^{\sqrt {\ell }-1} \left (\frac {np}{\ell }\right )^k \leq \frac {1}{\left (1-\frac {np}{\ell }\right )\left (1-\frac {1}{\sqrt {\ell }}\right )^{\sqrt {\ell }}} = O(1), \end{align*}
when
$\ell \geq (np)^4$
. Combining both the bounds along with Stirling’s approximation, we conclude that there is some
$C \gt 0$
such that
\begin{equation*} \mathbb{P}(X_\ell \in T^* | V_r) \leq \frac {r}{n^\ell }\left (\frac {e^{np}(np)^{\sqrt {\ell }}}{\sqrt {\ell }!} + O(1)\right ) \leq \frac {r}{n^\ell }\left (\frac {e^{np}(np)^{(np)^2}e^{(np)^2}}{(np)^{2(np)^2}}+O(1)\right ) \leq \frac {Cr}{n^\ell },\end{equation*}
when
$\ell \geq (np)^4$
and
$np \to \infty$
. Proceeding exactly as we did for the first inequality,
\begin{align*} \mathbb{P}(v_1,\ldots ,v_q \in T^*) &= \mathbb{P}(v_1 \in T^*) \cdot \mathbb{P}(v_2 \in T^* | V_1) \cdots \mathbb{P}(v_q \in T^* | V_{q-1}) \\ &\leq \frac {p^\ell }{\ell !}\frac {(q-1)!C^{q-1}}{n^{\ell (q-1)}} \leq \frac {(np)^{(np)^4}}{(np)^4!}\frac {(q-1)!C^{q-1}}{n^{\ell q}} = O\left ( \frac {(q-1)!C^{q-1}}{n^{\ell q}}\right ), \end{align*}
where in the final bound we use the fact that
$\frac {x^{x^4}}{(x^4)!} = o(1)$
as
$x\to \infty$
, which is an immediate consequence of Stirling’s approximation.
We may use Lemma 3 to bound the moments of the number of vertices reachable in a particular generation. For
$\ell \geq 0$
, denote by
$Z_\ell$
the number of vertices in
$T$
reachable from the root in the
$\ell$
-th generation.
Corollary 4.
For all integers
$\ell \ge 0$
and
$q \ge 1$
,
$\mathbb{E}[Z_\ell ^q] \leq (q -1)!e^{npq}$
. Furthermore, when
$\ell \geq (np)^4$
and
$np \to \infty$
, there is a constant
$C \gt 0$
such that
$\mathbb{E}[Z_\ell ^q] = O((q-1)!C^{q-1})$
.
Proof.
Denoting by
$S_\ell$
the set of
$n^{\ell }$
vertices in the
$\ell$
-th generation of
$T$
, we may write
$Z_\ell = \sum _{v\in S_{\ell }} \mathbf{1}_{\{v\in T^*\}}$
. Then
\begin{equation*} \mathbb{E} [Z_\ell ^q] = n^{\ell q}\mathbb{P}( v_1,\ldots ,v_q \in T^*)\,, \end{equation*}
where
$v_1,\ldots ,v_q$
are independent vertices chosen uniformly at random from
$S_\ell$
. Combining this with Lemma 3 implies the stated bounds.
The next bound will control the number of vertices that a typical vertex in a simple random temporal graph can reach, further allowing us to control the size of temporal cliques.
Theorem 5.
Let
$T^*$
be the set of vertices in
$T$
that are reachable from the root and suppose that
$np \to \infty$
. Then, for any integer
$q \geq 1$
, there is a constant
$c(q)$
such that
\begin{equation*}\mathbb{E}[|T^*|^q] \leq c(q)(np)^{4q}e^{npq}.\end{equation*}
Proof. Observe that
\begin{align*} \mathbb{E}|T^*|^q & = \mathbb{E}\left (\sum _{i=0}^{(np)^4}Z_i + \sum _{i \gt (np)^4} Z_i\right )^q \\ & \le 2^{q-1} \mathbb{E}\left (\sum _{i=0}^{(np)^4}Z_i \right )^q + 2^{q-1} \mathbb{E}\left (\sum _{i \gt (np)^4} Z_i\right )^q \quad \text{(by Jensen's inequality)} \\ & \leq \underbrace {2^{q-1}((np)^4+1)^{q-1}\mathbb{E}\left [ \sum _{i=0}^{(np)^4} Z_i^q \right ]}_{:= I} + \underbrace {2^{q-1}q\sum _{t \gt 0}t^{q-1}\mathbb{P}\left (\sum _{i \gt (np)^4} Z_i \geq t\right )}_{:= II}\,, \end{align*}
where in the last step we used Jensen’s inequality to bound the first term and the identity
$\mathbb{E}[X^q] = \int qt^{q-1}\mathbb{P}(X \gt t) dt$
to bound the second. The first inequality of Corollary 4 may be used to bound the expectation in term
$I$
, as
\begin{equation*}\mathbb{E}\left [ \sum _{i=0}^{(np)^4} Z_i^q\right ] \leq ((np)^{4}+1) \left (\sup _{i \geq 0}\mathbb{E}[Z_i^q]\right ) \leq ((np)^{4}+1) (q - 1)!e^{npq}.\end{equation*}
To bound
$II$
, we may write
\begin{align*} t^{q-1}\mathbb{P}\left ( \sum _{i \gt (np)^4} Z_i \geq t\right ) &\leq t^{q-1}\mathbb{P}\left ( Z_{(np)^4 + \log t} \gt 0\right ) + t^{q-1}\mathbb{P}\left (\bigcup _{i=(np)^4 + 1}^{(np)^4 + \log t} \left \{Z_i \geq \frac {t}{\log t}\right \}\right ) \\ &\leq \underbrace {t^{q-1}\mathbb{P}\left ( Z_{(np)^4 + \log t} \gt 0\right )}_{:=III} + \underbrace {t^{q-1}(\log t)\max _{(np)^4 \lt i \leq (np)^4 + \log t} \mathbb{P}\left (Z_i \geq \frac {t}{\log t}\right )}_{:= IV}, \end{align*}
and we can bound the two terms separately. To bound
$III$
, note that at level
$(np)^4+\log t$
, there are
$n^{(np)^4 + \log t}$
vertices, and they are each reachable with probability
$p^{(np)^4 + \log t}/((np)^4+ \log t)!$
. Thus, by Stirling’s approximation,
\begin{align*} III &\leq \frac {t^{q-1}(np)^{(np)^4 + \log t}}{((np)^4 + \log t)!} \\ &\leq \frac {t^{q-1}(enp)^{(np)^4 + \log t}}{(np)^{4(np)^4 + 4\log t}} \\ & = t^{q-1}\left ( \frac {e}{(np)^3} \right )^{(np)^4 + \log t} \\ &= t^{q-3\log (np)}\left (\frac {e}{(np)^3}\right )^{(np)^4} \end{align*}
for any
$t \geq 0$
. In particular, this implies that
$III$
is summable and converges to 0 when
$np \to \infty$
. Applying Markov’s inequality and the second inequality in Corollary 4 gives
\begin{equation*}IV \leq O\left ( \frac {\log ^{k+1} (t) ( k -1 )! C^{k-1} }{t^{k-q+1}}\right ),\end{equation*}
for any positive integer
$k$
and
$t \geq 0$
. Choosing
$k = q+1$
results in
$IV$
being summable and bounded above by a constant depending only on
$q$
. Grouping up all that only depends on
$q$
and upper bounding by some dominating constant
$c(q)$
we get
\begin{equation*}\mathbb{E}|T^*|^q \leq I + III + IV \leq c(q)(np)^{4q}e^{npq}.\end{equation*}
3. Proof of Theorem 1
We are now prepared to prove Theorem 1. For labelled vertices
$\{1,\ldots ,m\}$
to form a temporal clique in an RSTG they need to all be reachable from one another. This can happen if and only if for all distinct
$u,v \in \{1,\ldots ,m\}$
, there is a vertex
$w$
that can reach
$v$
with only edges that have time stamps above
$p/2$
, and is reachable from
$u$
with only edges that have time stamps below
$p/2$
. With this in mind, for any
$0\le a \lt b\le p$
, we define
$G_{[a,b]}$
to be the subgraph obtained from
$G$
by only keeping edges with time stamps in
$[a,b]$
. Set
$A_1,\ldots ,A_m$
to be the collection of all vertices that are reachable from
$1,\ldots ,m$
in
$G_{[0,p/2]}$
and
$B_1,\ldots ,B_m$
to be the collection of all vertices in
$G_{[p/2,p]}$
that can reach
$1,\ldots ,m$
. With this new notation, we can say that
$\{1,\ldots ,m\}$
form a temporal clique if for all
$i,j \in \{1,\ldots ,m\}$
distinct, the set
$A_i \cap B_j$
is nonempty.
It is important to note that
$G_{[p/2,p]}$
and
$G_{[0,p/2]}$
are identically distributed RSTGs, but are not independent. Observe that for any
$0\le a \lt b \le p$
, the RSTG
$G_{[a,b]}$
is determined by the binary vector
$X = (X_1,\ldots ,X_{\binom {n}{2}})$
defined by
$X_i = \mathbf{1}_{e_i \in G_{[a,b]}}$
(for some enumeration of the edges of
$K_n$
) and a random permutation
$O_{[a,b]}$
of
$[\binom {n}{2}]$
that denotes the relative orderings of the edge labels. In the next lemma we consider certain functionals of
$G_{[a,b]}$
, represented by
$X$
and
$O_{[a,b]}$
. More precisely, such a functional is of the form
$f\,:\,\{0,1\}^{\binom {n}{2}}\times \mathrm{Sym}(\binom {n}{2}) \to {\mathbb{R}}$
, where
$\mathrm{Sym}(\binom {n}{2})$
is the set of permutations of
$[\binom {n}{2}]$
. The next lemma deals with the dependence between two subgraphs
$G_{[a,b]}$
and
$G_{[c,d]}$
, for some
$0 \leq a \lt b \lt c \lt d \leq p$
.
Lemma 6.
Let
$G$
be an RSTG with vertices labelled
$\{1,\ldots ,n\}$
, and let
$0 \leq a \lt b \lt c \lt d \leq p$
. Set
$X_i = \mathbf{1}_{e_i \in G_{[a,b]}}$
,
$Y_i = \mathbf{1}_{e_i \in G_{[c,d]}}$
for some enumeration of the edges of
$K_n$
,
$e_1,\ldots ,e_{\binom {n}{2}}$
, and let
$X = (X_1,\ldots ,X_{\binom {n}{2}})$
and
$Y = (Y_1,\ldots ,Y_{\binom {n}{2}})$
. Let
$O_{[a,b]}$
and
$O_{[c,d]}$
be the permutations that denote the relative orderings of the edges in the two graphs. Let
$f,g\,:\,\{0,1\}^{\binom {n}{2}}\times \mathrm{Sym}(\binom {n}{2}) \to {\mathbb{R}}$
be such that
$f(x_1,\ldots ,x_{\binom {n}{2}},s)$
and
$g(x_1,\ldots ,x_{\binom {n}{2}},s)$
are two non-decreasing functions in
$x_1,\ldots ,x_{\binom {n}{2}}$
for any fixed
$s\in \mathrm{Sym}(\binom {n}{2})$
. Then
\begin{equation*}\mathbb{E}\left [f\left (X,O_{[a,b]}\right )g\left (Y,O_{[c,d]}\right )\right ] \leq \mathbb{E}\left [f\left (X,O_{[a,b]}\right )\right ] \mathbb{E}\left [g\left (Y,O_{[c,d]}\right )\right ].\end{equation*}
In particular,
\begin{align*}\mathbb{E}\left [ |A_1|^q \cdots |A_m|^q \cdot |B_1|^q \cdots |B_m|^q \right ]& \leq \mathbb{E}\left [|A_1|^q \cdots |A_m|^q \right ]\mathbb{E}\left [ |B_1|^q \cdots |B_m|^q\right ] \\& = \mathbb{E}\left [|A_1|^q \cdots |A_m|^q\right ]^2,\end{align*}
for all
$q \geq 0$
and
$A_1,B_1,\ldots ,A_m,B_m$
defined as above.
Proof.
Conditioned on
$Z= (X_1,Y_1,\ldots ,X_{\binom {n}{2}},Y_{\binom {n}{2}})$
, all of the randomness of
$f\left (X,O_{[a,b]}\right )$
and
$g\left (Y,O_{[c,d]}\right )$
comes from the random relative orderings. Since
$a \lt b \lt c \lt d$
, the two random variables
$O_{[a,b]}$
and
$O_{[c,d]}$
are independent, which implies that
$f\left (X,O_{[a,b]}\right )$
and
$g\left (Y,O_{[c,d]}\right )$
must also be conditionally independent on
$Z$
. Hence, by the tower property of conditional expectation,
\begin{align*} \mathbb{E}\left [f\left (X,O_{[a,b]}\right )g\left (Y,O_{[c,d]}\right )\right ] &= \mathbb{E}\left [\mathbb{E}\left [f\left (X,O_{[a,b]}\right )g\left (Y,O_{[c,d]}\right ) \Big | Z \right ]\right ] \\ &= \mathbb{E}\left [\mathbb{E}\left [f\left (X,O_{[a,b]}\right ) \Big | Z \right ]\mathbb{E}\left [g\left (Y,O_{[c,d]}\right ) \Big | Z \right ]\right ]. \\ &= \mathbb{E}\left [\mathbb{E}\left [f\left (X,O_{[a,b]}\right ) \Big | X \right ]\mathbb{E}\left [g\left (Y,O_{[c,d]}\right ) \Big | Y\right ]\right ], \end{align*}
where the final equality just follows from the fact that, once we condition on
$X$
, knowing
$Y$
tells us nothing about
$f(X,O_{[a,b]})$
and vice versa. The random variables
\begin{equation*}\mathbb{E}\left [f\left (X,O_{[a,b]}\right ) \Big | X = (z_1,\ldots ,z_{\binom {n}{2}}) \right ], \ \text{and} \ \mathbb{E}\left [g\left (Y,O_{[c,d]}\right ) \Big | Y = (z_1,\ldots ,z_{\binom {n}{2}}) \right ]\end{equation*}
are non-decreasing functions in
$z_1,\ldots ,z_{\binom {n}{2}}$
by the definitions of
$f$
and
$g$
. Furthermore, the collection of random variables
$\{X_i \,:\, i \in \{1,\ldots ,\binom {n}{2}\}\} \cup \{Y_i \,:\, i \in \{1,\ldots ,\binom {n}{2}\}\}$
are negatively associated (this can be seen by combining Proposition 7 and Lemma 8 from Dubhashi and Ranjan [Reference Dubhashi and Ranjan7]). With this, applying the tower property again gives
\begin{align*} \mathbb{E}\left [f\left (X,O_{[a,b]}\right )g\left (Y,O_{[c,d]}\right )\right ] \leq \mathbb{E}\left [f\left (X,O_{[a,b]}\right )\right ]\mathbb{E}\left [g\left (Y,O_{[c,d]}\right )\right ]. \end{align*}
Observing that
$|A_1| \cdots |A_m|$
and
$|B_1| \cdots |B_m|$
satisfy the conditions of the first statement and are identically distributed is enough to complete the proof of the second inequality.
The next lemma acts as a bridge between RSTGs and the temporal branching processes explored in the previous section. The idea behind the proof hinges on the fact that the sizes of
${\operatorname {binomial}}(n,p)$
branching processes upper bounds the sizes of neighbourhoods around vertices in an Erdős-Rényi graph, though formalising this idea takes some work. Equipped with this and Theorem5, the proof of Theorem1 is reduced to a routine use of the first-moment method.
Lemma 7.
$|A_1|$
is stochastically dominated by
$|T^*|$
, where
$T^*$
is the set of vertices reachable from the root in a temporal branching process
$T$
with offspring distribution
${\operatorname {binomial}}(n,p/2)$
. In particular,
$\mathbb{E}[|A_1|^q] \leq \mathbb{E}[|T^*|^q]$
for all
$q \geq 0$
.
Proof.
We can determine
$A_1$
via the foremost tree algorithm from Casteigts, Raskin, Renken, and Zamaraev [Reference Casteigts, Raskin, Renken and Zamaraev5]. The algorithm builds a tree recursively, building a tree of increasing paths starting from an arbitrary vertex. The algorithm is defined as follows:
-
• Initialise with
$\tau _0 = 1$
and
$G_0$
as the single vertex labelled
$1$
. -
• While
$\tau _k \leq p/2$
, set
$\tau _{k+1}$
to be the smallest time stamp of an edge connecting vertices in
$G_k$
with vertices outside of
$G_k$
that is larger than
$\tau _k$
. -
• If
$\tau _{k+1} \leq p/2$
, add the corresponding edge
$e_{k+1}$
and vertex
$v_{k+1}$
to obtain
$G_{k+1}$
. -
• If
$\tau _k\gt p/2$
, the algorithm terminates and outputs
$G_{k}$
.
Note that
$|A_1|$
equals the number of vertices of the resulting tree
$G_k$
, that is,
\begin{equation} |A_1| = \inf \left \{k \geq 0 \,:\, \tau _k \gt \frac {p}{2}\right \}\,. \end{equation}
This foremost tree algorithm can also be run on the tree
$T$
as a way to generate
$T^*$
with the same procedure, and we denote the sequence of timestamps in this graph as
$\tau ^*_k$
. Additionally, we denote by
$E_k$
and
$E^*_k$
the collection of all viable edges that could be added during step
$k$
, that is, all edges from
$G_k$
to
$K_n$
that, if added, keep the graph
$G_{k+1}$
as an increasing tree (all vertices reachable from the root). Note that by the definition of the algorithm, every edge in
$E_k$
must have a time stamp that is at least
$\tau _k$
and similarly for
$E_k^*$
and
$\tau ^*_k$
. Moreover, the time stamps of edges in
$E_k$
and
$E^*_k$
are uniformly distributed on
$[\tau _{k-1},1]$
and
$[\tau _{k-1}^*,1]$
, respectively. By means of a direct inductive coupling we show that
$\tau$
stochastically dominates
$\tau ^*$
,
$|E_k^*|$
stochastically dominates
$|E_k|$
, and hence
$|T^*|$
must stochastically dominate
$|A_1|$
by the characterisation of (2).
The base case of the induction is easy to see. By definition
$|E_1| = n-1$
,
$|E_1^*| = n$
,
$\tau _1 \sim \min _{1 \leq i \leq n-1} U_{1,i}$
, and
$\tau _1^* \sim \min _{1 \leq n} U_{1,i}$
. Thus, just using the same uniforms to generate both
$\tau _1$
and
$\tau _1^*$
is enough. Now suppose that there is some probability space
$(\Omega _{k-1},\mathcal{F}_{k-1},\mathbb{P}_{k-1})$
and random variables distributed as
$|E_{k-1}|,|E_{k-1}^*|,\tau _{k-1},\tau ^*_{k-1}$
(we just use the same symbols to denote these random variables) such that
$|E_{k-1}|(\omega ) \leq |E_{k-1}^*|(\omega )$
and
$\tau ^*_{k-1}(\omega ) \leq \tau _{k-1}(\omega )$
for all
$\omega \in \Omega _{k-1}$
. In the
$(k-1)$
-th step of the algorithm we added a new vertex to both graphs, resulting in
$n-k$
possible new edges to
$G$
and
$n$
edges to
$T$
for the
$k$
-th step. Hence, since we cannot add edges that are below
$\tau _{k-1}$
and
$\tau _{k-1}^*$
respectively
\begin{equation*}|E_k| \sim |E_{k-1}| + {\operatorname {binomial}}(n-k,1-\tau _{k-1}), \quad |E_k^*| \sim |E_{k-1}^*| + {\operatorname {binomial}}(n,1-\tau _k^*),\end{equation*}
and, recalling the distribution of time stamps in
$E_k$
and
$E_k^*$
,
\begin{equation*}\tau _k \sim \tau _{k-1} + (1-\tau _{k-1})\min _{1 \leq i \leq |E_k|} U_{k,i}, \quad \tau _k^* \sim \tau ^*_{k-1} + (1-\tau ^*_{k-1})\min _{1 \leq i \leq |E_k^*|} U_{k,i}.\end{equation*}
Let
$(\Omega _k, \mathcal{F}_k,\mathbb{P}_k)$
be the product of
$(\Omega _{k-1},\mathcal{F}_{k-1},\mathbb{P}_{k-1})$
with
$(\Omega ', \mathcal{F}',\mathbb{P}')$
, the probability space of
$\binom {n}{2}+n$
independent uniform random variables,
$(U_{k,i} \,:\, 1 \leq i \leq \binom {n}{2}+n)$
. Here we can couple the binomial random variables by generating them as
$\sum _{i=1}^k \mathbf{1}_{\{U_{k,i} \leq (1-\tau _{k-1})\}}$
and
$\sum _{i=1}^k \mathbf{1}_{\{U_{k,i} \leq (1-\tau ^*_{k-1}\}}$
respectively. Then, if we generate
$|E_k|$
and
$|E_k^*|$
with these binomials, by the inductive hypothesis, it must be the case that
$|E_k|(\omega ) \leq |E_k^*|(\omega )$
for all
$\omega \in \Omega _k$
. Similarly, using the uniforms
$(U_{k,i} \,:\, n+1 \leq i \leq |E_k^*|)$
to generate both
$\tau _k$
and
$\tau _k^*$
according to their distributions results in also having
$\tau _k^*(\omega ) \leq \tau _k(\omega )$
for all
$\omega \in \Omega _k$
.
With the lemmas out of the way we now prove our main result.
Proof of Theorem 1. Let
$m \geq 0$
and let
$A_1,\ldots ,A_m,B_1,\ldots ,B_m$
be as defined in the beginning of this section. Let
$N$
be the number of temporal cliques of size
$m$
in
$G$
. Then, if we take
$(v_{ij})_{i,j = 1}^m$
to be independently and uniformly chosen random vertices from the labelled set
$\{1,\ldots ,n\}$
, we may apply Lemma6 to get that
\begin{align*} \mathbb{E}[N] &\leq n^m\mathbb{P}(\{1,\ldots ,m\} \text{ form a temporal clique}) \\ &= n^{m+m(m-1)}\mathbb{P}(v_{ij} \in A_i, v_{ij} \in B_j \ \forall i \ne j) \\ &= n^{m+m(m-1)}\mathbb{E}\left [ \mathbb{P}\Big (v_{ij} \in A_i, v_{ij} \in B_j \ \forall i \ne j \Big | |A_1|,\ldots ,|A_m|,|B_1|,\ldots ,|B_m| \Big ) \right ] \\ &\leq n^{m+m(m-1)}\mathbb{E}\left [ \left ( \frac {|A_1|}{n} \right )^{m-1} \cdots \left ( \frac {|A_m|}{n} \right )^{m-1} \left ( \frac {|B_1|}{n} \right )^{m-1} \cdots \left ( \frac {|B_m|}{n} \right )^{m-1} \right ] \\ &\leq n^{m+m(m-1)}\mathbb{E}\left [ \left ( \frac {|A_1|}{n} \right )^{m-1} \cdots \left ( \frac {|A_m|}{n} \right )^{m-1}\right ]^2 \\ &\leq n^{m-m(m-1)}\mathbb{E}[|A_1|^{m-1} \cdots |A_m|^{m-1}]^2. \end{align*}
Applying Hölder’s inequality along with Lemma 7 and Theorem5 applied for the probability
$p/2 = c \log(n)/(2n)$
, gives us the upper bound
\begin{align*} \mathbb{E}[N] & \leq n^{m-m(m-1)}\mathbb{E}[|A_1|^{m(m-1)}]^2 \leq \kappa _m \left (\frac {c}{2}\log (n)\right )^{8m(m-1)}n^{m-m(m-1)+c(m(m-1))} \end{align*}
where
$\kappa _m$
is a constant depending on
$m$
only. If
$m-m(m-1)+cm(m-1) \lt 0$
, then
$\mathbb{E}[N] \to 0$
as
$n \to \infty$
. Rearranging, this inequality is equivalent to
$m \geq \lceil \frac {1}{1-c}+1\rceil$
as
$m$
is an integer.
Acknowledgements
Luc Devroye acknowledges the support of NSERC grant A3450. Gábor Lugosi acknowledges the support of AyudasFundación BBVA a Proyectos de Investigación Científica 2021 and the Spanish Ministry of Economy and Competitivenessgrant PID2022-138268NB-I00, financed by MCIN/AEI/10.13039/501100011033, FSE+MTM2015-67304-P, andFEDER, EU.).
